arXiv subjects

Emil A. Stoltenberg

Publications and source records attributed to Emil A. Stoltenberg.

3 recordsLinked to original sources

arXiv2026

Posterior uncertainty for kernel density estimates

Recent work in predictive Bayesian inference has enabled novel Bayesian interpretations of many well-known stochastic one-step-ahead predictive algorithms. In this paper, we study classic kernel density estimation in the predictive Bayesian framework. We prove that their predictive measures converge weakly almost surely $\unicode{x2013}$ meaning that their associated predictive resampling sequences are almost surely asymptotically exchangeable $\unicode{x2013}$ and we provide estimators for moments of the limiting random probability measure. We also show that the resampling sequences do not satisfy standard assumptions like being conditionally identically distributed (c.i.d.) or almost c.i.d. (a.c.i.d.), thus providing a non-trivial example of a predictive sequence which is not a.c.i.d. but nevertheless converges weakly almost surely. For Gaussian kernels, we show that the limiting directing measure is almost surely absolutely continuous with respect to the Lebesgue measure, meaning it emits a probability density. This enables us to derive credibility intervals for kernel density estimates, which we illustrate on two real datasets.

stat.ME↗

arXiv2025

Semiparametrics via parametrics and contiguity

Inference on the parametric part of a semiparametric model is no trivial task. If one approximates the infinite dimensional part of the semiparametric model by a parametric function, one obtains a parametric model that is in some sense close to the semiparametric model and inference may proceed by the method of maximum likelihood. Under regularity conditions, the ensuing maximum likelihood estimator is asymptotically normal and efficient in the approximating parametric model. Thus one obtains a sequence of asymptotically normal and efficient estimators in a sequence of growing parametric models that approximate the semiparametric model and, intuitively, the limiting 'semiparametric' estimator should be asymptotically normal and efficient as well. In this paper we make this intuition rigorous: we move much of the semiparametric analysis back into classical parametric terrain, and then translate our parametric results back to the semiparametric world by way of contiguity. Our approach departs from the conventional sieve literature by being more specific about the approximating parametric models, by working not only with but also under these when treating the parametric models, and by taking full advantage of the mutual contiguity that we require between the parametric and semiparametric models. We illustrate our theory with two canonical examples of semiparametric models, namely the partially linear regression model and the Cox regression model. An upshot of our theory is a new, relatively simple, and rather parametric proof of the efficiency of the Cox partial likelihood estimator.

math.ST↗

arXiv2020

A CLT for second difference estimators with an application to volatility and intensity

In this paper we introduce a general method for estimating the quadratic covariation of one or more spot parameters processes associated with continuous time semimartingales. This estimator is applicable to a wide range of spot parameter processes, and may also be used to estimate the leverage effect of stochastic volatility models. The estimator we introduce is based on sums of squared increments of second differences of the observed process, and the intervals over which the differences are computed are rolling and overlapping. This latter feature lets us take full advantage of the data, and, by sufficiency considerations, ought to outperform estimators that are only based on one partition of the observational window. The main result of the paper is a central limit theorem for such triangular array rolling quadratic variations. We highlight the wide applicability of this theorem by showcasing how it might be applied to a novel leverage effect estimator. The principal motivation for the present study, however, is that the discrete times at which a continuous time semimartingale is observed might depend on features of the observable process other than its level, such as its (non-observable) spot-volatility process. As the main application of our estimator, we therefore show how it may be used to estimate the quadratic covariation between the spot-volatility process and the intensity process of the observation times, when both of these are taken to be semimartingales. The finite sample properties of this estimator are studied by way of a simulation experiment, and we also apply this estimator in an empirical analysis of the Apple stock. Our analysis of the Apple stock indicates a rather strong correlation between the spot volatility process of the log-prices process and the times at which this stock is traded (hence observed).

math.ST↗